Lesson 6

There are other, more unexpected, consequences of Euler's formula.

Theorem (Sylvester–Gallai). For any \(n \in \mathbb{Z}_{\ge 3}\) points not lying on the same line, there exists a line containing exactly two points.

In the example below, each dashed line contains at least three points, and each solid one exactly two.

Theorem (Pick). The area of a simple polygon with vertices on the grid \(\mathbb{Z}^{2}\) is \[n_{\operatorname{int}}+\frac{n_{\operatorname{bd}}}{2}-1 \ ,\] where \(n_{\operatorname{int}}, n_{\operatorname{bd}}\) are the number of grid points inside and on the boundary of the polygon, respectively.

For example, the area of the polygon shown below is \(\textcolor{#ff5349}{4}+{\frac{\textcolor{myblue}{10}}{2}}-1=8\).